For the proof, we build on an observation due to Figueira. For
\(n \ge 0\) , let \(_{n \times n}\)  denote the \(n \times n\)  grid
database, defined in the obvious way with role name \(r_x\)  used for
horizontal edges and \(r_y\)  for vertical edges. It is noted in
[1]} that for one can find a database
\(\widehat{}_{n \times n}\)  of treewidth 2 such that the models of
\(\widehat{}_{n \times n}\)  and \(\lbrace {func}(r_x)\rbrace \)  are precisely the
models of \(_{n \times n}\)  and the size of \(_n\)  is polynomial
in \(n\) . We display the database \(\widehat{}_{n \times n}\)  in
Figure REF . Note that,
when the UNA is made, that database is simply unsatisfiable w.r.t. \(\lbrace {func}(r_x)\rbrace \) .
